The hyperbolic plane is a weird place. It's so big, that we can't fit the whole thing into 3D Euclidean space. The best we can do is draw weird, distorted pictures of it. But although each picture is distorted, we can combine different pictures which faithfully represent different aspects of the hyperbolic plane to build some intuition about what is going on. Three standard pictures are the halfspace model, the Poincare disk model, and the Klein model.
One strange feature of the hyperbolic plane, is that there are interesting triangles with infinitely long sides.